1: *> \brief \b DGEEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGEEQUB + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgeequb.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
22: * INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER INFO, LDA, M, N
26: * DOUBLE PRECISION AMAX, COLCND, ROWCND
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DGEEQUB computes row and column scalings intended to equilibrate an
39: *> M-by-N matrix A and reduce its condition number. R returns the row
40: *> scale factors and C the column scale factors, chosen to try to make
41: *> the largest element in each row and column of the matrix B with
42: *> elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
43: *> the radix.
44: *>
45: *> R(i) and C(j) are restricted to be a power of the radix between
46: *> SMLNUM = smallest safe number and BIGNUM = largest safe number. Use
47: *> of these scaling factors is not guaranteed to reduce the condition
48: *> number of A but works well in practice.
49: *>
50: *> This routine differs from DGEEQU by restricting the scaling factors
51: *> to a power of the radix. Baring over- and underflow, scaling by
52: *> these factors introduces no additional rounding errors. However, the
53: *> scaled entries' magnitured are no longer approximately 1 but lie
54: *> between sqrt(radix) and 1/sqrt(radix).
55: *> \endverbatim
56: *
57: * Arguments:
58: * ==========
59: *
60: *> \param[in] M
61: *> \verbatim
62: *> M is INTEGER
63: *> The number of rows of the matrix A. M >= 0.
64: *> \endverbatim
65: *>
66: *> \param[in] N
67: *> \verbatim
68: *> N is INTEGER
69: *> The number of columns of the matrix A. N >= 0.
70: *> \endverbatim
71: *>
72: *> \param[in] A
73: *> \verbatim
74: *> A is DOUBLE PRECISION array, dimension (LDA,N)
75: *> The M-by-N matrix whose equilibration factors are
76: *> to be computed.
77: *> \endverbatim
78: *>
79: *> \param[in] LDA
80: *> \verbatim
81: *> LDA is INTEGER
82: *> The leading dimension of the array A. LDA >= max(1,M).
83: *> \endverbatim
84: *>
85: *> \param[out] R
86: *> \verbatim
87: *> R is DOUBLE PRECISION array, dimension (M)
88: *> If INFO = 0 or INFO > M, R contains the row scale factors
89: *> for A.
90: *> \endverbatim
91: *>
92: *> \param[out] C
93: *> \verbatim
94: *> C is DOUBLE PRECISION array, dimension (N)
95: *> If INFO = 0, C contains the column scale factors for A.
96: *> \endverbatim
97: *>
98: *> \param[out] ROWCND
99: *> \verbatim
100: *> ROWCND is DOUBLE PRECISION
101: *> If INFO = 0 or INFO > M, ROWCND contains the ratio of the
102: *> smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
103: *> AMAX is neither too large nor too small, it is not worth
104: *> scaling by R.
105: *> \endverbatim
106: *>
107: *> \param[out] COLCND
108: *> \verbatim
109: *> COLCND is DOUBLE PRECISION
110: *> If INFO = 0, COLCND contains the ratio of the smallest
111: *> C(i) to the largest C(i). If COLCND >= 0.1, it is not
112: *> worth scaling by C.
113: *> \endverbatim
114: *>
115: *> \param[out] AMAX
116: *> \verbatim
117: *> AMAX is DOUBLE PRECISION
118: *> Absolute value of largest matrix element. If AMAX is very
119: *> close to overflow or very close to underflow, the matrix
120: *> should be scaled.
121: *> \endverbatim
122: *>
123: *> \param[out] INFO
124: *> \verbatim
125: *> INFO is INTEGER
126: *> = 0: successful exit
127: *> < 0: if INFO = -i, the i-th argument had an illegal value
128: *> > 0: if INFO = i, and i is
129: *> <= M: the i-th row of A is exactly zero
130: *> > M: the (i-M)-th column of A is exactly zero
131: *> \endverbatim
132: *
133: * Authors:
134: * ========
135: *
136: *> \author Univ. of Tennessee
137: *> \author Univ. of California Berkeley
138: *> \author Univ. of Colorado Denver
139: *> \author NAG Ltd.
140: *
141: *> \date November 2011
142: *
143: *> \ingroup doubleGEcomputational
144: *
145: * =====================================================================
146: SUBROUTINE DGEEQUB( M, N, A, LDA, R, C, ROWCND, COLCND, AMAX,
147: $ INFO )
148: *
149: * -- LAPACK computational routine (version 3.4.0) --
150: * -- LAPACK is a software package provided by Univ. of Tennessee, --
151: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
152: * November 2011
153: *
154: * .. Scalar Arguments ..
155: INTEGER INFO, LDA, M, N
156: DOUBLE PRECISION AMAX, COLCND, ROWCND
157: * ..
158: * .. Array Arguments ..
159: DOUBLE PRECISION A( LDA, * ), C( * ), R( * )
160: * ..
161: *
162: * =====================================================================
163: *
164: * .. Parameters ..
165: DOUBLE PRECISION ONE, ZERO
166: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
167: * ..
168: * .. Local Scalars ..
169: INTEGER I, J
170: DOUBLE PRECISION BIGNUM, RCMAX, RCMIN, SMLNUM, RADIX, LOGRDX
171: * ..
172: * .. External Functions ..
173: DOUBLE PRECISION DLAMCH
174: EXTERNAL DLAMCH
175: * ..
176: * .. External Subroutines ..
177: EXTERNAL XERBLA
178: * ..
179: * .. Intrinsic Functions ..
180: INTRINSIC ABS, MAX, MIN, LOG
181: * ..
182: * .. Executable Statements ..
183: *
184: * Test the input parameters.
185: *
186: INFO = 0
187: IF( M.LT.0 ) THEN
188: INFO = -1
189: ELSE IF( N.LT.0 ) THEN
190: INFO = -2
191: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
192: INFO = -4
193: END IF
194: IF( INFO.NE.0 ) THEN
195: CALL XERBLA( 'DGEEQUB', -INFO )
196: RETURN
197: END IF
198: *
199: * Quick return if possible.
200: *
201: IF( M.EQ.0 .OR. N.EQ.0 ) THEN
202: ROWCND = ONE
203: COLCND = ONE
204: AMAX = ZERO
205: RETURN
206: END IF
207: *
208: * Get machine constants. Assume SMLNUM is a power of the radix.
209: *
210: SMLNUM = DLAMCH( 'S' )
211: BIGNUM = ONE / SMLNUM
212: RADIX = DLAMCH( 'B' )
213: LOGRDX = LOG( RADIX )
214: *
215: * Compute row scale factors.
216: *
217: DO 10 I = 1, M
218: R( I ) = ZERO
219: 10 CONTINUE
220: *
221: * Find the maximum element in each row.
222: *
223: DO 30 J = 1, N
224: DO 20 I = 1, M
225: R( I ) = MAX( R( I ), ABS( A( I, J ) ) )
226: 20 CONTINUE
227: 30 CONTINUE
228: DO I = 1, M
229: IF( R( I ).GT.ZERO ) THEN
230: R( I ) = RADIX**INT( LOG( R( I ) ) / LOGRDX )
231: END IF
232: END DO
233: *
234: * Find the maximum and minimum scale factors.
235: *
236: RCMIN = BIGNUM
237: RCMAX = ZERO
238: DO 40 I = 1, M
239: RCMAX = MAX( RCMAX, R( I ) )
240: RCMIN = MIN( RCMIN, R( I ) )
241: 40 CONTINUE
242: AMAX = RCMAX
243: *
244: IF( RCMIN.EQ.ZERO ) THEN
245: *
246: * Find the first zero scale factor and return an error code.
247: *
248: DO 50 I = 1, M
249: IF( R( I ).EQ.ZERO ) THEN
250: INFO = I
251: RETURN
252: END IF
253: 50 CONTINUE
254: ELSE
255: *
256: * Invert the scale factors.
257: *
258: DO 60 I = 1, M
259: R( I ) = ONE / MIN( MAX( R( I ), SMLNUM ), BIGNUM )
260: 60 CONTINUE
261: *
262: * Compute ROWCND = min(R(I)) / max(R(I)).
263: *
264: ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
265: END IF
266: *
267: * Compute column scale factors
268: *
269: DO 70 J = 1, N
270: C( J ) = ZERO
271: 70 CONTINUE
272: *
273: * Find the maximum element in each column,
274: * assuming the row scaling computed above.
275: *
276: DO 90 J = 1, N
277: DO 80 I = 1, M
278: C( J ) = MAX( C( J ), ABS( A( I, J ) )*R( I ) )
279: 80 CONTINUE
280: IF( C( J ).GT.ZERO ) THEN
281: C( J ) = RADIX**INT( LOG( C( J ) ) / LOGRDX )
282: END IF
283: 90 CONTINUE
284: *
285: * Find the maximum and minimum scale factors.
286: *
287: RCMIN = BIGNUM
288: RCMAX = ZERO
289: DO 100 J = 1, N
290: RCMIN = MIN( RCMIN, C( J ) )
291: RCMAX = MAX( RCMAX, C( J ) )
292: 100 CONTINUE
293: *
294: IF( RCMIN.EQ.ZERO ) THEN
295: *
296: * Find the first zero scale factor and return an error code.
297: *
298: DO 110 J = 1, N
299: IF( C( J ).EQ.ZERO ) THEN
300: INFO = M + J
301: RETURN
302: END IF
303: 110 CONTINUE
304: ELSE
305: *
306: * Invert the scale factors.
307: *
308: DO 120 J = 1, N
309: C( J ) = ONE / MIN( MAX( C( J ), SMLNUM ), BIGNUM )
310: 120 CONTINUE
311: *
312: * Compute COLCND = min(C(J)) / max(C(J)).
313: *
314: COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
315: END IF
316: *
317: RETURN
318: *
319: * End of DGEEQUB
320: *
321: END
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